Re: Cantor's circular "proof" that evens = integers
- From: george <greeneg@xxxxxxxxxx>
- Date: Sat, 02 Jun 2007 08:08:11 -0700
On May 28, 9:14 am, stevendaryl3...@xxxxxxxxx (Daryl McCullough)
wrote:
What is to be established is that the formula Proof(x,w)
is true in the intended domain if and only if x is a code
for a formula and w is a code for the proof of that formula.
This is *not* established by a proof in the object theory.
Of course not; it can't be, because proofs and formulas
have to be finite, but the object theory doesn't understand
finitude. The object theory does not know about itself
that some of its models are going to have elements s
with the property that infinitely many statements of the
form n<s are true. Because of the compactness theorem,
the theory can't use that infinitary conjunction as a criterion
for declaring such s "infinite". More to the point, since
we are talking about the THEORY, the theory is not aware
of the existence of ANYthing infinite, since, by the completeness
theorem, the theory, since it consists only of the provable
sentences, contains only the sentences true in ALL models,
and, therefore, contains only sentences true in the standard
model, where NOTHING infinite exists.
It *can't* be, since the object theory doesn't mention proofs
or formulas.
THAT is NOT relevant! The problem is NOT what the object
theory does or doesn't MENTION! You can DEFINE and/or
ENCODE things that you don't mention! You can define
and encode proofs and formulas! The problem is that because
they are philosophically REQUIRED to have this characteristic
that is NOT fully first-order definable, you can't define them
fully accurately!
If the results are provable then it won't
matter what they are about.
I don't know what you are talking about here.
If what results are provable?
Any wff or sentence can be a result, but we are most interested
in sentences using the concepts were talking about here.
For concreteness, let's expand the signature by adding one defined
predicate, Prf(p,f), which is true
iff p encodes a proof of the formula encoded by f.
Prf is not syntactically eliminable any more than + or * is,
but it is, presumably, like them, "simplifiable". Some wffs
that make non-trivial use of this predicate are provable.
In particular, however, "results" of the form Ap[~Prf(p,x)]
are NOT provable. Still, the point is, you NEED to be
QUANTIFYING OVER the whole domain UNDER "Prf"
IN ORDER to have this problem. The rest of the time,
Prf CAN be "tractable".
..The problem you face is that these results ARE NOT provable.
The problem you face is that ANY AND EVERY 1st-order
predicate you ATTEMPT to define as Prf(p,f) IS ABSOLUTELY
GUARANTEED TO GET IT *WRONG* in SOME model
Yes, that's why I said that Prf(p,f) *only* formalizes proof
in some models.
IF you have formalized it RIGHT then it OUGHT
to come up right in all models. If it turns out you
can't do that, then A DIFFERENT APPROACH IS
INDICATED.
It doesn't formalize proof in nonstandard models.
COME *ON*! You are ONLY looking at the WFF!
You are ONLY looking at the STRING! You are ONLY
LOOKING at a sentence in the language of PA, or PA extended
by a few carefully defined notions (like a proof-predicate)!
JUST LOOKING AT THE SENTENCE does NOT REVEAL
ANYTHING ABOUT *ANY* MODEL!
MORE to the point, if the sentence IS PROVABLE, then
IT DOESN'T MATTER what model you intended it to be
interpreted in! So you canNOT BEGIN by saying "be sure
to interpret this sentence in my intended model"! If you don't
have a proof of the sentence in the object theory, WHY
SHOULD ANYONE EVEN *BELIEVE* the sentence AT ALL?
Why should anyone even BOTHER TO TRY "interpreting" it??
If you don't have a proof then we know in advance that it is FALSE
at least SOME of the time! Sure, we COULD go in search of
the needle (the standard model) in the haystack (of all models
of the axioms), but WHY even BOTHER? THERE ARE PLENTY
OF OTHER RICH RESULTS AND RESEARCH OPPORTUNITIES
*back* among the theorems that WERE provable!
More to the point, to say of anything that "it means this in
this model but not in that one" is To Do VIOLENCE to the
MEANING of "meaning". Meaning is flexible but NOT to THAT
degree. "Today" will not MEAN something different tomorrow
from what it meant yesterday,
DESPITE the fact that it refers to different
days in the different contexts. Its MEANING is THE SAME.
"Odd" and "Even" mean THE SAME thing in standard and non-
standard models. PRIME means the SAME thing in standard
and non-standard models. GUESS WHAT: ALL *meanings*,
EVEN the ones that are indexical enough to be model-SENSITIVE,
are STILL model-INDEPENDENT, BY DEFINITION.
So if some nonstandard model satisfies the formula
~Con(PA)
it isn't correct to say that that "PA is inconsistent" is true
in that model, because Con(PA) doesn't mean consistency
in that model.
If it doesn't mean "PA is consistent" in that model then it
doesn't mean it in any other model either. MEANING is NOT
what varies among the models. But let's look again at the
language levels: you just turned these phrases:
"PA is inconsistent" is true in that model
Please!
You said "It isn't correct to say that" but the point is,
saying THAT isn't even GRAMMATICAL, let ALONE
"correct"! The reason WHY it isn't correct to say this is
NOT that the truth-values clash! It IS that "PA is consistent"
CAN'T be evaluated in that model or structure because IT
IS IN THE WRONG LANGUAGE! It is at the wrong level!
It is IN NATURAL language! It is in the meta-language!
The models/interpretations/structures
assign truth-values to sentences IN THE FORMAL languages
depending on how the interpretation has interpreted their FORMAL
components! "In that model", we evaluate sentences from
the OBJECT language, NOT the meta-language!
.
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